boolean expression simplificationgajski/eecs31/slides/digital_design_-_boolean... · principles of...
TRANSCRIPT
Principles Of Digital Design
Boolean Expression Simplification
Literal Minimization Boolean Factoring Implicants and Prime Implicants K Maps Don’t Care Conditions
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine
Expression Minimization
Goal? Literal minimization Ex: xy + xy’ = x (y + y’) = x Ex: xyz + xyz’ +xy’z + xy’z’ = x (yz + yz’ + y’z + y’z’) = x
How? Visualization (few variables) K- Maps (2-6 variables)
Reality? CAD tools (many variables) Ex: 64-bit adder (64 variables, 2 terms)
2
64
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 3
Boolean Functions and Factors Each Boolean function of n variables can be represented
by a truth table where each raw represents a minterm Each subset of n-m literals, l1l2…ln – m , is called a factor iff
l1l2…ln – m (any minterm of m variables) is a 1-minterm
x y z F 0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
x y z F 0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
y’z = (x’+ x) y’z
xy = xy(z’ + z)
z’ = (x’y’ + x’y + xy’ + xy)z
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 4
Boolean Functions and Factors For any Boolean function a prime implicant is a factor not
contained in any other prime implicant An essential prime implicant is a factor that contains a 1-
minterm that is not included in any other prime implicant
x y z F 0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
x y z F 0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
y’z = (x’+ x) y’z
xy = xy(z’ + z)
z = (x’y’ + x’y + xy’ + xy)z
F = xy + z
x y z F 0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 1
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 5
K-maps define Boolean functions
Map representation is equivalent to truth tables, Boolean expressions
Maps aid in visually identifying prime implicants and essential prime implicants in each Boolean function
Maps are used for manual optimization of Boolean functions with few variables
Map Representation
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 6
m0 m1
m2 m3
Karnaugh Maps for n = 1, 2, 3, and 4
n = 1 n = 2
n = 3 n = 4
1
0
1 0
m0 m1 m3 m2
m4 m5 m7 m6
m12 m13 m15 m14
m8 m9 m11 m10
m0 m1 m3 m2
m4 m5 m7 m6
10 11 01 00
10 11 01 00
1
0
01
00
10
11
x
y
yz x
x
zw xy
1 0
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 7
0 1
2 3
2–variable K-Map
1 0
1
0
y
x 0 1
2 3
1 0
1
0
y
x
0 1
2 3
1 0
1
0
y
x 0 1
2 3
1 0
1
0
y
x 0 1
2 3
1 0
1
0
y
x
x′y′
1 1 1
1
1
1
x y AND OR XOR
0 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1 0
x′y
xy′ xy
Truth Table
AND OR XOR
Map Organization Example of 1-factors
Example:
Factor x
Factor y
Factor x′
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 8
Three–variable K-Map
1
0 0 1 3 2
4 5 7 6
10 11 01 00 yz
x
x′y′z′ x′y′z x′yz x′yz′
xy′z′ xy′z xyz xyz′
Map Organization
1
0 0 1 3 2
4 5 7 6
10 11 01 00 yz
x
Example of 2-factors
1
0 0 1 3 2
4 5 7 6
10 11 01 00 yz
x
Example of 1-factors
Factor xz′
Factor yz
Factor x′y′
Factor x
Factor z′
Factor z
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 9
Maps for Carry and Sum Functions
ci xi yi ci + 1 si 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1
Truth Table Carry Function ci + 1 Sum Function si
1
0 0 1 3 2
4 5 7 6
10 11 01 00 xiyi
ci
1
1 1 1 1
0 0 1 3 2
4 5 7 6
10 11 01 00 xiyi
ci
1 1
1 1
si = xi′yi′ci + xi′yici′ + xiyi′ci′ + xiyici ci + 1 = xiyi + cixi + ciyi
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 10
Four–variable K-Map
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
zw xy
x′y′z′w′ x′y′z′w x′y′zw x′y′zw′
x′yz′w′ x′yz′w x′yzw x′yzw′
xyz′w′ xyz′w xyzw xyzw′
xy′z′w′ xy′z′w xy′zw xy′zw′
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
zw xy
Factor w′
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
zw xy
Factor x′
Factor y′w
Factor x′y
Factor xz
Map Organization
Example of 3-factors Example of 2-factors
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 11
Simplification of Comparison Functions
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
1 1 1
1 1
1
Less-than Function
x1 x0 y1 y0 Greater
Than Equal Less Than
0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 0 1 0 1 1 0 0 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
y1y0 x1x0
1
1 1 1
1 1
Greater-than Function
Truth Table
G = x1y1′ + x0y1′y0′ + x1x0y0′
y1y0 x1x0
L = x1′ y1 + x1′x0′y0 + x0′y1y0
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 12
Boolean Simplification with Maps Truth table, canonical form
or standard form
Determine prime implicants
Generate map
Select essential prime implicants
Find minimal cover
Standard form
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 13
Boolean Simplification with Maps Example: Maps method Problem: Using the map method, simplify the Boolean function
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
yz wx
1 1 1
1 1
1 1
1 1 1
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
yz
wx
1 1 1
1 1
1 1
1 1 1
F = w′y′z′ + wz + xyz + w′y
Map Organization
PI List: w′z′, wz, yz, w′y EPI List: w′z′, wz Cover List: (1) w′z′, wz, yz (2) w′z′, wz, w′y
Prime Implicants in the Map
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 14
Boolean Simplification with Maps Example: Map method Problem: Simplify the Boolean function
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
yz wx
1 1
1 1
1 1
1 1
F = w′x′yz′ + w′xy + wxz + wx′y′ + w′x′y′z′
PI List: w′x′z′, w′xy, wxz, wx′y′, x′y′z′, wy′z, xyz, w′yz′ EPI List: empty Cover List: (1) w′x′z′, w′xy, wxz, wx′y′ (2) x′y′z′, wy′z, xyz, w′yz′
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 15
Don’t–Care Conditions
Completely specified functions have a value assigned for every minterm
Incompletely specified functions do not have values assigned for some minterms which are called don’t–care minterms (d–minterms) or don’t–care conditions
Don’t–care minterms can be assigned any value during simplifications in order to simplify Boolean expressions
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 16
Don’t–Care Conditions Example: Don’t–care conditions
Problem: Derive Boolean expressions for the 9’s complement of a BCD digit
Digits Nine’s Complements
Decimal BCD
Decimal BCD
x3 x2 x1 x0 y3 y2 y1 y0
0 0 0 0 0 9 1 0 0 1 1 0 0 0 1 8 1 0 0 0 2 0 0 1 0 7 0 1 1 1 3 0 0 1 1 6 0 1 1 0 4 0 1 0 0 5 0 1 0 1 5 0 1 0 1 4 0 1 0 0 6 0 1 1 0 3 0 0 1 1 7 0 1 1 1 2 0 0 1 0 8 1 0 0 0 1 0 0 0 1 9 1 0 0 1 0 0 0 0 0
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
x1x0
x3x2
1
X
1
X X
X
X
X
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
x1x0
x3x2
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
x1x0
x3x2
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
10 11 01 00
01
00
10
11
x1x0
x3x2
X X X
X
X
X
X X X
X
X
X
X X X
X
X
X
1 1
1 1
1 1
1 1
1
1
1
1
1
y2 = x2 ⊕ x1 y3 = x3′ x2′ x1′
y0 = x0′ y1 = x1 K Maps Nine’s–Complement Table
Copyright © 2010-2011 by Daniel D. Gajski EECS 31/CSE 31, University of California, Irvine 17
Summary
Simplification of Boolean expressions by K-Map method (visual for 2-6 variables))
Prime implicants
Essential prime implicants
Minimal cover